Can we consolidate the definition of “critical point” in one-variable calculus? Instead of saying that a critical point is a point x such that $f’(x) = 0$ or $f$ is not differentiable at $x$, why not just say that it is a point $x$ such that $\dfrac{1}{f’(x)}$ fails to be defined?

We could do that, but would doing so gain us anything except increased obfuscation?